Electro-dynamic loudspeaker principle
Exemple : See the practical work on the speaker.
Fondamental : Mechanic and electric equations of the loudspeaker
The following figure represents a device used as a speaker or microphone.
is a permanent magnet of revolution symmetry around axis.
In the air gap is a radial magnetic field.
In the area where the coil moves, attached to the pavilion , the magnetic field is written :
is a system of mass , likely to move along axis.
It can be set in translation movement if it is subject to an exterior force .
It is also subject to dissipative forces of sum , spring forces of sum applied by a system of springs, as well as Laplace's forces of sum applied on .
The pavilion has a total length of wire equal to and carries an intensity .
It is admitted, by neglecting the helicity of , that each element of the wire can be represented in cylindrical coordinates by :
is powered by a voltage source through the circuit.
, and are the total resistance, inductance and capacitance, relatively to the whole of the circuit, included.
can be expressed as a function of , and .
Indeed, if we sum Laplace's forces on several elements of :
The inertia center theorem applied to the pavilion and projected on axis gives the differential equation (M) verified by .
It shows the mechanical behavior of the system :
The mechanic equation (M) of the system is obtained.
In order to obtain the electric equation, , and the induced electromotive force in must be expressed as functions of .
The electromotive force is induced on an element of :
After integration along in the positive direction of :
(We also can obtain this expression for using an analogy with the Laplace rails).
Kirchhoff's law gives the electric equation (E) of the circuit :
Fondamental : Energy review
Let's compute :
This equation can be written as such :
represents the mechanic energy of the system and its electromagnetic energy.
The previous equation expresses the energy conservation :
The derivative of is equal to the sum of powers provided by the two energy sources of the system : the exterior force and the voltage source .
To this is subtracted the power dissipated via friction forces (or acoustic energy when the speaker emits a noise) or Joule effect.
Méthode : Study in forced sinusoidal regime
The system is now under a sinusoidal steady state at fixed frequency.
The equations (M') and (E') linking the complex representations of , , and can be written as such :
The complex notation in has been introduced and :
is the complex impedance of the series dipole.
is the mechanic impedance of the mobile pavilion (ratio between Laplace's force and speed).
When it is used as a speaker, is equal to zero, the energy of the system comes from the source of voltage .
The answer to the system is characterized by relations and , given by (E') and (M') :
The quantity is introduced :
From an electro kinetic point of view, it is as though, because of the movement in the magnetic field, another electric impedance is added to .
This impedance, called motion impedance, characterizes the electro-mechanic coupling done by the assemblage.
If the frequency is included in , the vibrations of the pavilion will create an acoustic pressure wave which will generate audible sound.
When it is used as a microphone, then is equal to zero.
The energy of the system comes from the sinusoidal force which expresses the action of the pressure forces from a sound wave on .
The answer of the system is then characterized by the function :
The intensity is proportional to the applied force : the electric signal received is faithful to the force.
It will be recorded, treated and reproduced by another speaker.